Adaptive Moving Mesh Research Articles

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

Well-balanced positivity preserving adaptive moving mesh central-upwind schemes for the Saint-Venant system

We extend the adaptive moving mesh (AMM) central-upwind schemes recently proposed in Kurganov et al. [Commun. Appl. Math. Comput. 3 (2021) 445–479] in the context of one- (1-D) and two-dimensional (2-D) Euler equations of gas dynamics and granular hydrodynamics, to the 1-D and 2-D Saint-Venant system of shallow water equations. When the bottom topography is nonflat, these equations form hyperbolic systems of balance laws, for which a good numerical method should be capable of preserving a delicate balance between the flux and source terms as well as preserving the nonnegativity of water depth even in the presence of dry or almost dry regions. Therefore, in order to extend the AMM central-upwind schemes to the Saint-Venant systems, we develop special positivity preserving reconstruction and evolution steps of the AMM algorithms as well as special corrections of the solution projection step in (almost) dry areas. At the same time, we enforce the moving mesh to be structured even in the case of complicated 2-D computational domains. We test the designed method on a number of 1-D and 2-D examples that demonstrate robustness and high resolution of the proposed numerical approach.

Solving the Schrödinger-Poisson system using the coordinate adaptive moving mesh method

In this paper, we implement the Adaptive Moving Mesh method (AMM) to the solution of initial value problems involving the Schr\"odinger equation, and more specifically the Schr\"odinger-Poisson system of equations. This method is based on the solution of the problem on a discrete domain, whose resolution is coordinate and time-dependent, and allows to dynamically assign numerical resolution in terms of desired refinement criteria. We apply the method to solve various test problems involving stationary solutions of the SP system, and toy scenarios related to the disruption of subhalo s made of ultralight bosonic dark matter traveling on top of host galaxies.

An Adaptive Moving Mesh Method for the Five-Equation Model

An Adaptive Moving Mesh Method for the Five-Equation Model

Adaptive Moving Mesh Finite Volume Method Experimentation to Solving Riemann's Type Hyperbolic Problems

Adaptive Moving Mesh Finite Volume Method Experimentation to Solving Riemann's Type Hyperbolic Problems

Adaptive Moving Mesh Central-Upwind Schemes for Hyperbolic System of PDEs: Applications to Compressible Euler Equations and Granular Hydrodynamics

We introduce adaptive moving mesh central-upwind schemes for one- and two-dimensional hyperbolic systems of conservation and balance laws. The proposed methods consist of three steps. First, the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh. When the evolution step is complete, the grid points are redistributed according to the moving mesh differential equation. Finally, the evolved solution is projected onto the new mesh in a conservative manner. The resulting adaptive moving mesh methods are applied to the one- and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems. Our numerical results demonstrate that in both cases, the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.

An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical examples.

Adaptive moving mesh upwind scheme for the two-species chemotaxis model

Chemotaxis systems are used to model the propagation, aggregation and pattern formation of bacteria/cells in response to an external stimulus, usually a chemical one. A common property of all chemotaxis systems is their ability to model a concentration phenomenon—rapid growth of the cell density in small neighborhoods of concentration points/curves. More precisely, the solution may develop singular, spiky structures, or even blow up in finite time. Therefore, the development of accurate and computationally efficient numerical methods for the chemotaxis models is a challenging task.We study the two-species Patlak–Keller–Segel type chemotaxis system, in which the two species do not compete, but have different chemotactic sensitivities, which may lead to a significantly difference in cell density growth rates. This phenomenon was numerically investigated in Kurganov and Lukáčová-Medviďová (2014) and Chertock et al. (2018), where second- and higher-order methods on uniform Cartesian grids were developed. However, in order to achieve high resolution of the density spikes developed by the species with a lower chemotactic sensitivity, a very fine mesh had to be utilized and thus the efficiency of the numerical method was affected.In this work, we consider an alternative approach relying on mesh adaptation, which helps to improve the approximation of the singular structures evolved by chemotaxis models. We develop, in particular, an adaptive moving mesh (AMM) finite-volume semi-discrete upwind method for the two-species chemotaxis system. The proposed AMM technique allows one to increase the density of mesh nodes at the blowup regions. This helps to substantially improve the resolution while using a relatively small number of finite-volume cells.

An Adaptive Moving Mesh Method for Forced Curve Shortening Flow.

We propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation. Control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle, where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve. Central finite differences are used to discretize in space the governing evolution equation for the position vector, and a second-order implicit scheme is used for the temporal integration. Simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second-order accuracy. Furthermore, the new method delivers improved solution accuracy compared to the use of uniform arc-length meshes.

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a $C^0$ finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative termuxxplays a more important role thanuxxxxin dissipation.

Adaptive Moving Mesh Finite Element Method for Space Fractional Advection Dispersion Equation

Space fractional advection dispersion equation includes non-local differential operators, which leads to calculating numerical integrals with weakly singular kernel on every subintervals. Oscillations are visible in computed solutions when fractional order tends to 1. Studies show that the stiffness matrix of time semi-discretization can be calculated directly by formulas established from a special variational formulation. Oscillations are eliminated by using adaptive moving mesh and De Boor algorithm, while the number of nodes remains unchanged.

Book Reviews

Readers will be lucky to find two featured reviews in this issue, praising their subjects. One, by Rabi Bhattacharya, is about Fischer's History of the Central Limit Theorem. The other, by Alfio Quarteroni, reviews Huang and Russell's Adaptive Moving Mesh Methods. The history of the CLT involves lots of prominent mathematicians over a period of two hundred years, mostly before asymptotics became ordinary. The other is central to current efforts to computationally solve practical problems involving partial differential equations using adaptive meshes. Other reviews are about computer arithmetic, infinite-dimensional dynamical systems, chaos, population dynamics, and other topics. They illustrate the continuing breadth and vitality of current publications in applied math, in our increasingly electronic environment. We hope you enjoy reading them.

A High‐Resolution Adaptive Moving Mesh Hydrodynamic Algorithm

An algorithm for simulating self-gravitating cosmological astrophysical fluids is presented. The advantages include a large dynamic range, parallelizability, high resolution per grid element, and fast execution speed. The code is based on a finite volume flux-conservative total variation diminishing (TVD) scheme for the shock-capturing hydro and an iterative multigrid solver for the gravity. The grid is a time-dependent field, whose motion is described by a generalized potential flow. Approximately constant mass per cell can be obtained, which provides all the advantages of a Lagrangian scheme. The grid deformation combined with appropriate limiting and smoothing schemes guarantees a regular and well-behaved grid geometry, in which nearest neighbor relationships remain constant. The full hydrodynamic fluid equations are implemented in the curvilinear moving grid, which allows for arbitrary fluid flow relative to the grid geometry. This combination retains all the advantages of the grid-based schemes including high speed per fluid element and a rapid gravity solver. The current implementation is described, and empirical simulation results are presented. Accurate execution speed calculations are given in terms of floating point operations per time step per grid cell. This code is freely available to the community.